Grade: 7 Unit: 4 Week: 4 Content: Circumference of a Circle Area of a Circle Dates: 2/11 – 2/15
Theme Essential Question: How can you use formulas and geometry to solve real-world and mathematical problems?
Essential Questions:
How do you apply the circumference and area of a circle to solve problems?
What is the relationship between the circumference and area of a circle?
Standards:
7.G. 4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Objectives:
The student will explore, examine, and formulate viable arguments for the development of the formulas for circumference and area of a circle.
The student will apply the formulas for the circumference and area of a circle to real world and mathematical problems.
The student will provide a viable argument supporting the relationship between circumference and area of a circle.
Reflections and/or Comments from your PCSSD 7th Grade Curriculum Team STOP: Carefully read the standard for this week’s lesson. Students are to know and then apply the formulas for circumference and area of a circle. They will also have to develop a derivation for the relationship between circumference and area. Read the lesson development in the OnCore book. Students are being asked to look deeper into the relationships pertaining to circles. It is going beyond just using C = 2πr or πd and A = πr2.
Background: Recommended: For a quick overview of the standard(s) to be addressed in this Unit, see Arizona’s Content Standards Reference Materials athttp://www.azed.gov/educator-certification/
Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2022
Development of circle concepts: This is the students’ initial work with circles. Knowing that a circle is created by connecting all the points equidistant from a point (center) is essential to understanding the relationships between radius, diameter, circumference, pi and area. Students can observe this by folding a paper plate several times, finding the center at the intersection, then measuring the lengths between the center and several points on the circle, the radius. Measuring the folds through the center, or diameters leads to the realization that a diameter is two times a radius. Circumference: Given multiple-size circles, students should then explore the relationship between the radius and the length measure of the circle (circumference) finding an approximation of pi and ultimately deriving a formula for circumference. String or yarn can be laid over the circle and compared to a ruler is an adequate estimate of the circumference. Recommended Activities: http://illuminations.nctm.org/LessonDetail.aspx?id=U159 Using estimation and measurement skills, students will determine the ratio of circumference to diameter and explore the meaning of π. Students will discover the circumference and area formulas based on their investigations. http://illuminations.nctm.org/ActivityDetail.aspx?ID=116 With this three-part online applet, students can explore with graphic and numeric displays how the circumference and area of a circle compare to its radius and diameter. Students can collect data points by dragging the radius to various lengths and clicking the "Add to Table" button to record the data in the table. http://illuminations.nctm.org/LessonDetail.aspx?id=L647 Using a MIRATM geometry tool, students determine the relationships between radius, diameter, circumference and area of a circle. http://www.utdanacenter.org/mathtoolkit/instruction/lessons/6_exploring.php Students measure circular objects to collect data to investigate the relationship between the circumference of a circle and its diameter. They find that, regardless of the size of the object or the size of the measuring unit, it always takes a little more than three times the length of the diameter to measure the circumference. Area: This same process can be followed in finding the relationship between the diameter and the area of a circle by using grid paper to estimate the area. Another visual for understanding the area of a circle can be modeled by cutting up a paper plate into 16 pieces along diameters and reshaping the pieces into a parallelogram (See OnCore book). In figuring area of a circle, the squaring of the radius can also be explained by showing a circle inside a square. Again, the formula is derived and then learned.
Recommended Activity: http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Circles_and_Their_Areas.pdf Given that units of area are squares, how can we find the area of a circle or other curved region? Imagine a waffle-like grid inside a circle and a larger grid containing the circle. The area of the circle lies between the area of the inside grid and the area of the outside grid.
After explorations, students should then solve problems, set in relevant contexts, using the formulas for area and circumference.
Assessment: Product: This product will be developed throughout the next four lessons, 4-4 to 4-6 and 5-1 with displays and/or presentation during 5-2.
Situation: ZOO TRAINER You and your partner have been hired by the zoo to be animal trainers. You will be required to develop a training and living facility for your animal. You will be receiving specifications for the design of your facility form the zoo manager.
You have been given the option to select your animal. It must be a land animal suitable for this task, such as a llama, goat, elephant, etc.
Phase I: Design Training corral
The training corral will be circular. You are to determine the dimensions that will be appropriate for training your animal. Two gates are to be place in your design:
Trainer entrance gate (4’ opening) and
Future explanations gate (6’ or larger depending on your animal).
Prepare a scale drawing of your proposal (leave room for future expansion). Be sure to provide a scale in the drawing design.
You are to use chain link fencing fabric 10’ high. How much fencing will be needed? Use the internet and provide an estimate for cost. Prepare detail documentation of your calculations.
The training corral will be covered with sod. Provide the square yards needed to cover the ground. Sod will arrive in 2’ by 1’ sections. How many sections will be needed? Prepare detail documentation of your calculations.
Key Questions
What is the formula for circumference of a circle and how is it informally developed?
What is the formula for the area of a circle and how is it informally developed?
What is the relationship between the circumference and the area of a circle and how is it informally developed?
Observable Student Behaviors (Performance)
The student can apply and use the formulas for circumference and area of a circle to real world and mathematical problems.
The student can give an informal derivation of the relationship between the circumference and the area of a circle.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary:
Math
Diameter Radius Circumference Chord Center Area Pi (π)
Suggested Activities:
Houghton Mifflin OnCore Mathematics Middle School Grade 7
Unit 5-1, Circumference of a Circle, p. 111-114
Unit 5-2, Area of a Circle, p. 115-118
ABC Mastering the Common Core in Mathematics
Chapter 12.1, Parts of a Circle, p. 149
Chapter 12.2, Circumference, p. 150-151
Chapter 12.3, Area of a Circle, p. 152-153
Chapter 12.4, Relating Circumference and Area, p. 154
Chapter 12.5, Circle Word Problems, p. 155
Chapter 12 Review and Test, p. 156-158
Gizmo Correlation:
7.G.4
Circles: Circumference and Area
Resize a circle and compare its radius, circumference, and area.
Teaching the Common Core Math Standards with Hands-On Activities,
7.G.4 – Activity#1, Circle Scavenger Hunt, p.121
The Circle Scavenger Hunt found in the Teaching the Common Core Math Standards with Hands-On Activities would be a great activity to work into this unit. It involves real-life mathematics items students have all around them. By planning ahead of time you could have students bring in different size circles, or as the activity suggests measure circles they find from home.
7.G.4 – Activity #2, What’s the Relationship?, p. 122 (Similar to OnCore p. 117)
Highly Recommended
http://illustrativemathematics.org/illustrations/34(7.G.4) The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
Glencoe 7th Grade Mathematics Application and Concepts Course 2, Chapter 5-1 and 5-2, p. 149-157
Mathematics in Children's Literature: Many children's books include math concepts and can be used to help teach them in a fun way. This website includes several annotated Lists of Children's Literature including the math concepts and grade levels.
The teaching Channel currently offers videos of K-12 mathematics teaching aligned with the Common Core Sate Standards, which would be perfect for professional development with teacher teams.
Interactivate is a set of free, online courseware for exploration in science and mathematics. It is comprised of activities, lessons, and discussions. The site is structured around collections of activities, lessons, and discussions.
Content: Circumference of a Circle
Area of a Circle
Dates: 2/11 – 2/15
Theme Essential Question:
How can you use formulas and geometry to solve real-world and mathematical problems?
Essential Questions:
Standards:
Objectives:
Reflections and/or Comments from your PCSSD 7th Grade Curriculum Team
STOP: Carefully read the standard for this week’s lesson. Students are to know and then apply the formulas for circumference and area of a circle. They will also have to develop a derivation for the relationship between circumference and area. Read the lesson development in the OnCore book. Students are being asked to look deeper into the relationships pertaining to circles. It is going beyond just using C = 2πr or πd and A = πr2.
Background:
Recommended: For a quick overview of the standard(s) to be addressed in this Unit, see Arizona’s Content Standards Reference Materials athttp://www.azed.gov/educator-certification/
Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2022
Development of circle concepts:
This is the students’ initial work with circles. Knowing that a circle is created by connecting all the points equidistant from a point (center) is essential to understanding the relationships between radius, diameter, circumference, pi and area. Students can observe this by folding a paper plate several times, finding the center at the intersection, then measuring the lengths between the center and several points on the circle, the radius. Measuring the folds through the center, or diameters leads to the realization that a diameter is two times a radius.
Circumference:
Given multiple-size circles, students should then explore the relationship between the radius and the length measure of the circle (circumference) finding an approximation of pi and ultimately deriving a formula for circumference. String or yarn can be laid over the circle and compared to a ruler is an adequate estimate of the circumference.
Recommended Activities:
http://illuminations.nctm.org/LessonDetail.aspx?id=U159
Using estimation and measurement skills, students will determine the ratio of circumference to diameter and explore the meaning of π. Students will discover the circumference and area formulas based on their investigations.
http://illuminations.nctm.org/ActivityDetail.aspx?ID=116
With this three-part online applet, students can explore with graphic and numeric displays how the circumference and area of a circle compare to its radius and diameter. Students can collect data points by dragging the radius to various lengths and clicking the "Add to Table" button to record the data in the table.
http://illuminations.nctm.org/LessonDetail.aspx?id=L647
Using a MIRATM geometry tool, students determine the relationships between radius, diameter, circumference and area of a circle.
http://www.utdanacenter.org/mathtoolkit/instruction/lessons/6_exploring.php
Students measure circular objects to collect data to investigate the relationship between the circumference of a circle and its diameter. They find that, regardless of the size of the object or the size of the measuring unit, it always takes a little more than three times the length of the diameter to measure the circumference.
Area:
This same process can be followed in finding the relationship between the diameter and the area of a circle by using grid paper to estimate the area.
Another visual for understanding the area of a circle can be modeled by cutting up a paper plate into 16 pieces along diameters and reshaping the pieces into a parallelogram (See OnCore book). In figuring area of a circle, the squaring of the radius can also be explained by showing a circle inside a square. Again, the formula is derived and then learned.
Recommended Activity:
http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Circles_and_Their_Areas.pdf
Given that units of area are squares, how can we find the area of a circle or other curved region? Imagine a waffle-like grid inside a circle and a larger grid containing the circle. The area of the circle lies between the area of the inside grid and the area of the outside grid.
After explorations, students should then solve problems, set in relevant contexts, using the formulas for area and circumference.
Assessment:
Product:
This product will be developed throughout the next four lessons, 4-4 to 4-6 and 5-1 with displays and/or presentation during 5-2.
Situation:
ZOO TRAINER
You and your partner have been hired by the zoo to be animal trainers. You will be required to develop a training and living facility for your animal. You will be receiving specifications for the design of your facility form the zoo manager.
- The training corral will be circular. You are to determine the dimensions that will be appropriate for training your animal. Two gates are to be place in your design:
- Trainer entrance gate (4’ opening) and
- Future explanations gate (6’ or larger depending on your animal).
Prepare a scale drawing of your proposal (leave room for future expansion). Be sure to provide a scale in the drawing design.Key Questions
Observable Student Behaviors (Performance)
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Vocabulary:
Center Area Pi (π)
Suggested Activities:
- Houghton Mifflin OnCore Mathematics Middle School Grade 7
- Unit 5-1, Circumference of a Circle, p. 111-114
- Unit 5-2, Area of a Circle, p. 115-118
- ABC Mastering the Common Core in Mathematics
- Chapter 12.1, Parts of a Circle, p. 149
- Chapter 12.2, Circumference, p. 150-151
- Chapter 12.3, Area of a Circle, p. 152-153
- Chapter 12.4, Relating Circumference and Area, p. 154
- Chapter 12.5, Circle Word Problems, p. 155
- Chapter 12 Review and Test, p. 156-158
- Gizmo Correlation:
- 7.G.4
- Circles: Circumference and Area
- Resize a circle and compare its radius, circumference, and area.
- Teaching the Common Core Math Standards with Hands-On Activities,
- 7.G.4 – Activity#1, Circle Scavenger Hunt, p.121
The Circle Scavenger Hunt found in the Teaching the Common Core Math Standards with Hands-On Activities would be a great activity to work into this unit. It involves real-life mathematics items students have all around them. By planning ahead of time you could have students bring in different size circles, or as the activity suggests measure circles they find from home.- 7.G.4 – Activity #2, What’s the Relationship?, p. 122 (Similar to OnCore p. 117)
- Highly Recommended
http://illustrativemathematics.org/illustrations/34 (7.G.4)The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
- Glencoe 7th Grade Mathematics Application and Concepts Course 2, Chapter 5-1 and 5-2, p. 149-157
- Create a puzzle with unit vocabulary words.
http://www.discoveryeducation.com/free-puzzlemaker/?CFID=1276695&CFTOKEN=75709576Diverse Learners
Homework: (Teachers Discretion)
Terminology for Teachers:
Ethnicity/Culture | Immigration/Migration | Intercultural Competence | Socialization | Racism/Discrimination
High Yield Strategies
Similarities/Differences | Summarizing/Notetaking | Reinforcing/Recognition | Homework/Practice |
Non-Linguistic representation| Cooperative Learning | Objectives/Feedback |
Generating-Testing Hypothesis | Cues, Questions, Organizers
Resources:
Professional Texts:
Literary Texts
- //Literature// Books - //Sir Conference// Series
- Geometry and Literature Titles:
http://sci.tamucc.edu/~eyoung/geometry_literature.html- Measurement and children’s Literature/Ratios
http://sci.tamucc.edu/~eyoung/measure_literature.html- Mathematics in Children's Literature: Many children's books include math concepts and can be used to help teach them in a fun way. This website includes several annotated Lists of Children's Literature including the math concepts and grade levels.
Click on the following link, http://libguides.nl.edu/mathinchildrenslit, and then look under Math and Literature Bibliographies.- Middle & High School: Literature in Mathematics
Many books include websites with lesson ideas.http://sci.tamucc.edu/~eyoung/middle_school_literature.html
- Lesson Plans for Using Literature in Middle and High School Mathematics (developed by Leonor and edited by Elaine)
http://sci.tamucc.edu/~eyoung/Literature%20webpages/Leonor/index.html- Miscellaneous Math and Children's Literature
http://sci.tamucc.edu/~eyoung/literature.htmlInformational Texts
Art, Music, and Media
Manipulatives:
- Grid paper
- Measuring tape
- Ruler
- Different size circle cut outs
- Strings
- Versa Tiles
http://www.learner.org/courses/learningmath/geometry/session9/part_c/index.html http://nlvm.usu.edu Geometry softwareGames
Videos
SMART Board Lessons, Promethean Lessons
- 7. G. 4Circumference, Volume, Surface Area
Students will practice finding the Circumference, Volume, and Surface Area of given shapes/figures.- 7. G. 4 Area of a Circle
Formula for area of a circle and how to use a grid and squares to check reasonableness or answer.Other Activities, etc.
- Puzzle Maker
- http://www.discoveryeducation.com/free-puzzlemaker/?CFID=1276695&CFTOKEN=75709576
- Interactivate is a set of free, online courseware for exploration in science and mathematics. It is comprised of activities, lessons, and discussions. The site is structured around collections of activities, lessons, and discussions.
http://www.shodor.org/interactivate/guide/Activities and tools:
http://www.shodor.org/interactivate/activities/
Language
Arts
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
7 Matrix
Accelerated 7
Matrix
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Home K-2
Home 3-6
Home 6-8
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6